Question

The figure shows the front view of a convex lens, which originally had only one edge. Five holes of different shapes, namely triangle, square, pentagon, hexagon and circle, were drilled through it...What is the total number of edges in the lens after the holes were drilled? 

 

Hint:

When analyzing changes to a 3D object, think about all three types of features: faces, edges, and vertices. Drilling a hole adds new inner faces, new edges where these faces meet each other and the original faces, and new vertices where the edges meet.

Answer 1

Step 1: Understanding the Concept: 
An edge in 3D geometry is the line segment where two faces of a solid object meet. We need to count the initial number of edges on the lens and then add the new edges created by drilling each of the five holes. 

Step 2: Detailed Explanation: 
Initial State: 
A convex lens has one original edge, which is the circle where the two curved surfaces meet. \(\textit{Initial Edges = 1} \)
Edges created by drilling a hole: 
When a prismatic hole with an n-sided polygon base is drilled through a solid: 
It creates `n` new edges on the front face. 
It creates `n` new edges on the back face. 
It creates `n` new longitudinal edges inside the hole, where the inner walls of the hole meet. 
So, a polygonal hole with `n` sides adds a total of \(n + n + n = 3n\) edges. A circular hole is like a cylinder. It has two circular edges (one on the front, one on the back) and no longitudinal edges. So, it adds 2 edges. 
Calculating Edges for Each Hole: 
Triangle (n=3): adds \(3 \times 3 = 9\) edges. 
Square (n=4): adds \(3 \times 4 = 12\) edges. 
Pentagon (n=5): adds \(3 \times 5 = 15\) edges. 
Hexagon (n=6): adds \(3 \times 6 = 18\) edges. 
Circle: adds 2 edges. 
Step 3: Final Answer: 
To find the total number of edges, we sum the original edge and all the newly created edges. \[ \text{Total Edges} = \text{Original Edge} + \text{Edges from Holes} \] \[ \text{Total Edges} = 1 + (\text{triangle}) + (\text{square}) + (\text{pentagon}) + (\text{hexagon}) + (\text{circle}) \] \[ \text{Total Edges} = 1 + 9 + 12 + 15 + 18 + 2 \] \[ \text{Total Edges} = 57 \] The total number of edges in the lens after drilling the holes is 57.